For numerical differentiation, dimensionality can be a blessing!

Blessing of Massive Scale: Spatial Graphical Model Estimation with a Total Cardinality Constraint

We consider the problem of estimating high dimensional spatial graphical models with a total cardinality constraint (i.e., the `0-constraint). Though this problem is highly nonconvex, we show that its primal-dual gap diminishes linearly with the dimensionality and provide a convex geometry justification of this ‘blessing of massive scale’ phenomenon. Motivated by this result, we propose an effi...

A Blessing of Dimensionality: Measure Concentration and Probabilistic Inference

Low Complexity Gaussian Latent Factor Models and a Blessing of Dimensionality

Learning the structure of graphical models from data usually incurs a heavy curse of dimensionality that renders this problem intractable in many real-world situations. The rare cases where the curse becomes a blessing provide insight into the limits of the efficiently computable and augment the scarce options for treating very under-sampled, high-dimensional data. We study a special class of G...

Embracing the Blessing of Dimensionality in Factor Models

Factor modeling is an essential tool for exploring intrinsic dependence structures among high-dimensional random variables. Much progress has been made for estimating the covariance matrix from a high-dimensional factor model. However, the blessing of dimensionality has not yet been fully embraced in the literature: much of the available data is often ignored in constructing covariance matrix e...

Blessing of dimensionality: mathematical foundations of the statistical physics of data

The concentrations of measure phenomena were discovered as the mathematical background to statistical mechanics at the end of the nineteenth/beginning of the twentieth century and have been explored in mathematics ever since. At the beginning of the twenty-first century, it became clear that the proper utilization of these phenomena in machine learning might transform the curse of dimensionalit...